3.3.45 \(\int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [245]

Optimal. Leaf size=306 \[ \frac {2 b (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {b f \text {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {b f \text {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {b^2 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {b^2 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2} \]

[Out]

2*b*(f*x+e)*arctanh(exp(d*x+c))/a^2/d-(f*x+e)*coth(d*x+c)/a/d+f*ln(sinh(d*x+c))/a/d^2+b*f*polylog(2,-exp(d*x+c
))/a^2/d^2-b*f*polylog(2,exp(d*x+c))/a^2/d^2+b^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/d/(a^2+b^2
)^(1/2)-b^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/d/(a^2+b^2)^(1/2)+b^2*f*polylog(2,-b*exp(d*x+c)
/(a-(a^2+b^2)^(1/2)))/a^2/d^2/(a^2+b^2)^(1/2)-b^2*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/d^2/(a^2+
b^2)^(1/2)

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Rubi [A]
time = 0.39, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5694, 4269, 3556, 4267, 2317, 2438, 3403, 2296, 2221} \begin {gather*} \frac {b^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2 \sqrt {a^2+b^2}}-\frac {b^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2 \sqrt {a^2+b^2}}+\frac {b^2 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^2 d \sqrt {a^2+b^2}}-\frac {b^2 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^2 d \sqrt {a^2+b^2}}+\frac {b f \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^2}-\frac {b f \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 b (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}+\frac {f \log (\sinh (c+d x))}{a d^2}-\frac {(e+f x) \coth (c+d x)}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e + f*x)*Csch[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]

[Out]

(2*b*(e + f*x)*ArcTanh[E^(c + d*x)])/(a^2*d) - ((e + f*x)*Coth[c + d*x])/(a*d) + (b^2*(e + f*x)*Log[1 + (b*E^(
c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) - (b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a
^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (f*Log[Sinh[c + d*x]])/(a*d^2) + (b*f*PolyLog[2, -E^(c + d*x)])/(a^2*d^
2) - (b*f*PolyLog[2, E^(c + d*x)])/(a^2*d^2) + (b^2*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a
^2*Sqrt[a^2 + b^2]*d^2) - (b^2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^
2)

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3403

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/((-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 5694

Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/a, Int[(e + f*x)^m*Csch[c + d*x]^n, x], x] - Dist[b/a, Int[(e + f*x)^m*(Csch[c + d*x]^(n - 1)/(
a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x) \text {csch}^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac {(e+f x) \coth (c+d x)}{a d}-\frac {b \int (e+f x) \text {csch}(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {e+f x}{a+b \sinh (c+d x)} \, dx}{a^2}+\frac {f \int \coth (c+d x) \, dx}{a d}\\ &=\frac {2 b (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {\left (2 b^2\right ) \int \frac {e^{c+d x} (e+f x)}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a^2}+\frac {(b f) \int \log \left (1-e^{c+d x}\right ) \, dx}{a^2 d}-\frac {(b f) \int \log \left (1+e^{c+d x}\right ) \, dx}{a^2 d}\\ &=\frac {2 b (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {\left (2 b^3\right ) \int \frac {e^{c+d x} (e+f x)}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a^2 \sqrt {a^2+b^2}}-\frac {\left (2 b^3\right ) \int \frac {e^{c+d x} (e+f x)}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a^2 \sqrt {a^2+b^2}}+\frac {(b f) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^2}-\frac {(b f) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^2}\\ &=\frac {2 b (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {b f \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^2}-\frac {b f \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^2}-\frac {\left (b^2 f\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 \sqrt {a^2+b^2} d}+\frac {\left (b^2 f\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 \sqrt {a^2+b^2} d}\\ &=\frac {2 b (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {b f \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^2}-\frac {b f \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^2}-\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 \sqrt {a^2+b^2} d^2}\\ &=\frac {2 b (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {b f \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^2}-\frac {b f \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^2}+\frac {b^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {b^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}\\ \end {align*}

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Mathematica [A]
time = 3.48, size = 405, normalized size = 1.32 \begin {gather*} \frac {-a d (e+f x) \coth \left (\frac {1}{2} (c+d x)\right )+2 a f \log (\sinh (c+d x))-2 b d e \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+2 b c f \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+2 b f \left (-\left ((c+d x) \left (\log \left (1-e^{-c-d x}\right )-\log \left (1+e^{-c-d x}\right )\right )\right )-\text {PolyLog}\left (2,-e^{-c-d x}\right )+\text {PolyLog}\left (2,e^{-c-d x}\right )\right )+\frac {2 b^2 \left (-2 d e \tanh ^{-1}\left (\frac {a+b \cosh (c+d x)+b \sinh (c+d x)}{\sqrt {a^2+b^2}}\right )+2 c f \tanh ^{-1}\left (\frac {a+b \cosh (c+d x)+b \sinh (c+d x)}{\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b (\cosh (c+d x)+\sinh (c+d x))}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b (\cosh (c+d x)+\sinh (c+d x))}{a+\sqrt {a^2+b^2}}\right )+f \text {PolyLog}\left (2,\frac {b (\cosh (c+d x)+\sinh (c+d x))}{-a+\sqrt {a^2+b^2}}\right )-f \text {PolyLog}\left (2,-\frac {b (\cosh (c+d x)+\sinh (c+d x))}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2}}-a d (e+f x) \tanh \left (\frac {1}{2} (c+d x)\right )}{2 a^2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((e + f*x)*Csch[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]

[Out]

(-(a*d*(e + f*x)*Coth[(c + d*x)/2]) + 2*a*f*Log[Sinh[c + d*x]] - 2*b*d*e*Log[Tanh[(c + d*x)/2]] + 2*b*c*f*Log[
Tanh[(c + d*x)/2]] + 2*b*f*(-((c + d*x)*(Log[1 - E^(-c - d*x)] - Log[1 + E^(-c - d*x)])) - PolyLog[2, -E^(-c -
d*x)] + PolyLog[2, E^(-c - d*x)]) + (2*b^2*(-2*d*e*ArcTanh[(a + b*Cosh[c + d*x] + b*Sinh[c + d*x])/Sqrt[a^2 +
b^2]] + 2*c*f*ArcTanh[(a + b*Cosh[c + d*x] + b*Sinh[c + d*x])/Sqrt[a^2 + b^2]] + f*(c + d*x)*Log[1 + (b*(Cosh
[c + d*x] + Sinh[c + d*x]))/(a - Sqrt[a^2 + b^2])] - f*(c + d*x)*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(
a + Sqrt[a^2 + b^2])] + f*PolyLog[2, (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(-a + Sqrt[a^2 + b^2])] - f*PolyLog[2
, -((b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2]))]))/Sqrt[a^2 + b^2] - a*d*(e + f*x)*Tanh[(c + d*
x)/2])/(2*a^2*d^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(625\) vs. \(2(283)=566\).
time = 1.57, size = 626, normalized size = 2.05

method	result	size
risch	\(-\frac {2 \left (f x +e \right )}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )}-\frac {2 f \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {f \ln \left ({\mathrm e}^{d x +c}+1\right )}{a \,d^{2}}+\frac {f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}+\frac {b e \ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d}-\frac {b e \ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d}-\frac {2 b^{2} e \arctanh \left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{2} d \sqrt {a^{2}+b^{2}}}+\frac {b f c \ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d^{2}}+\frac {2 b^{2} f c \arctanh \left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{2} d^{2} \sqrt {a^{2}+b^{2}}}+\frac {b f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{a^{2} d}+\frac {b f \dilog \left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d^{2}}+\frac {b f \dilog \left ({\mathrm e}^{d x +c}\right )}{a^{2} d^{2}}+\frac {b^{2} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{a^{2} d \sqrt {a^{2}+b^{2}}}+\frac {b^{2} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{a^{2} d^{2} \sqrt {a^{2}+b^{2}}}-\frac {b^{2} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{a^{2} d \sqrt {a^{2}+b^{2}}}-\frac {b^{2} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{a^{2} d^{2} \sqrt {a^{2}+b^{2}}}+\frac {b^{2} f \dilog \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{a^{2} d^{2} \sqrt {a^{2}+b^{2}}}-\frac {b^{2} f \dilog \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{a^{2} d^{2} \sqrt {a^{2}+b^{2}}}\)	\(626\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-2/d*(f*x+e)/a/(exp(2*d*x+2*c)-1)-2/a/d^2*f*ln(exp(d*x+c))+1/a/d^2*f*ln(exp(d*x+c)+1)+1/a/d^2*f*ln(exp(d*x+c)-
1)+1/a^2/d*b*e*ln(exp(d*x+c)+1)-1/a^2/d*b*e*ln(exp(d*x+c)-1)-2/a^2/d*b^2*e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*ex
p(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/a^2/d^2*b*f*c*ln(exp(d*x+c)-1)+2/a^2/d^2*b^2*f*c/(a^2+b^2)^(1/2)*arctanh(1/2*
(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/a^2/d*b*f*ln(exp(d*x+c)+1)*x+1/a^2/d^2*b*f*dilog(exp(d*x+c)+1)+1/a^2/d
^2*b*f*dilog(exp(d*x+c))+1/a^2/d*b^2*f/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2
)))*x+1/a^2/d^2*b^2*f/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/a^2/d*b^2
*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-1/a^2/d^2*b^2*f/(a^2+b^2)^(1/2)*
ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+1/a^2/d^2*b^2*f/(a^2+b^2)^(1/2)*dilog((-b*exp(d*x+c
)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/a^2/d^2*b^2*f/(a^2+b^2)^(1/2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)
+a)/(a+(a^2+b^2)^(1/2)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

(4*b^2*integrate(1/2*x*e^(d*x + c)/(a^2*b*e^(2*d*x + 2*c) + 2*a^3*e^(d*x + c) - a^2*b), x) - 4*b*d*integrate(1
/4*x/(a^2*d*e^(d*x + c) + a^2*d), x) - 4*b*d*integrate(1/4*x/(a^2*d*e^(d*x + c) - a^2*d), x) - a*((d*x + c)/(a
^2*d^2) - log(e^(d*x + c) + 1)/(a^2*d^2)) - a*((d*x + c)/(a^2*d^2) - log(e^(d*x + c) - 1)/(a^2*d^2)) - 2*x/(a*
d*e^(2*d*x + 2*c) - a*d))*f + (b^2*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 +
b^2)))/(sqrt(a^2 + b^2)*a^2*d) + b*log(e^(-d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c) - 1)/(a^2*d) + 2/((a*e^
(-2*d*x - 2*c) - a)*d))*e

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2027 vs. \(2 (283) = 566\).
time = 0.37, size = 2027, normalized size = 6.62 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

(2*(a^3 + a*b^2)*c*f - 2*(a^3 + a*b^2)*d*cosh(1) - 2*((a^3 + a*b^2)*d*f*x + (a^3 + a*b^2)*c*f)*cosh(d*x + c)^2
- 2*(a^3 + a*b^2)*d*sinh(1) - 4*((a^3 + a*b^2)*d*f*x + (a^3 + a*b^2)*c*f)*cosh(d*x + c)*sinh(d*x + c) - 2*((a
^3 + a*b^2)*d*f*x + (a^3 + a*b^2)*c*f)*sinh(d*x + c)^2 + (b^3*f*cosh(d*x + c)^2 + 2*b^3*f*cosh(d*x + c)*sinh(d
*x + c) + b^3*f*sinh(d*x + c)^2 - b^3*f)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*c
osh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - (b^3*f*cosh(d*x + c)^2 + 2*b^3*f*cosh(d*x
+ c)*sinh(d*x + c) + b^3*f*sinh(d*x + c)^2 - b^3*f)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x
+ c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - (b^3*c*f - b^3*d*cosh(1) - b^3*
d*sinh(1) - (b^3*c*f - b^3*d*cosh(1) - b^3*d*sinh(1))*cosh(d*x + c)^2 - 2*(b^3*c*f - b^3*d*cosh(1) - b^3*d*sin
h(1))*cosh(d*x + c)*sinh(d*x + c) - (b^3*c*f - b^3*d*cosh(1) - b^3*d*sinh(1))*sinh(d*x + c)^2)*sqrt((a^2 + b^2
)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + (b^3*c*f - b^3*d*cosh(1)
- b^3*d*sinh(1) - (b^3*c*f - b^3*d*cosh(1) - b^3*d*sinh(1))*cosh(d*x + c)^2 - 2*(b^3*c*f - b^3*d*cosh(1) - b^
3*d*sinh(1))*cosh(d*x + c)*sinh(d*x + c) - (b^3*c*f - b^3*d*cosh(1) - b^3*d*sinh(1))*sinh(d*x + c)^2)*sqrt((a^
2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - (b^3*d*f*x + b^3*
c*f - (b^3*d*f*x + b^3*c*f)*cosh(d*x + c)^2 - 2*(b^3*d*f*x + b^3*c*f)*cosh(d*x + c)*sinh(d*x + c) - (b^3*d*f*x
+ b^3*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c)
+ b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + (b^3*d*f*x + b^3*c*f - (b^3*d*f*x + b^3*c*f)*cosh(d*x + c)^
2 - 2*(b^3*d*f*x + b^3*c*f)*cosh(d*x + c)*sinh(d*x + c) - (b^3*d*f*x + b^3*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b
^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) -
b)/b) - ((a^2*b + b^3)*f*cosh(d*x + c)^2 + 2*(a^2*b + b^3)*f*cosh(d*x + c)*sinh(d*x + c) + (a^2*b + b^3)*f*si
nh(d*x + c)^2 - (a^2*b + b^3)*f)*dilog(cosh(d*x + c) + sinh(d*x + c)) + ((a^2*b + b^3)*f*cosh(d*x + c)^2 + 2*(
a^2*b + b^3)*f*cosh(d*x + c)*sinh(d*x + c) + (a^2*b + b^3)*f*sinh(d*x + c)^2 - (a^2*b + b^3)*f)*dilog(-cosh(d*
x + c) - sinh(d*x + c)) - ((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*cosh(1) - ((a^2*b + b^3)*d*f*x + (a^2*b + b^3
)*d*cosh(1) + (a^2*b + b^3)*d*sinh(1) + (a^3 + a*b^2)*f)*cosh(d*x + c)^2 + (a^2*b + b^3)*d*sinh(1) - 2*((a^2*b
+ b^3)*d*f*x + (a^2*b + b^3)*d*cosh(1) + (a^2*b + b^3)*d*sinh(1) + (a^3 + a*b^2)*f)*cosh(d*x + c)*sinh(d*x +
c) - ((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*cosh(1) + (a^2*b + b^3)*d*sinh(1) + (a^3 + a*b^2)*f)*sinh(d*x + c)
^2 + (a^3 + a*b^2)*f)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + ((a^2*b + b^3)*d*cosh(1) - ((a^2*b + b^3)*d*cos
h(1) + (a^2*b + b^3)*d*sinh(1) - (a^3 + a*b^2 + (a^2*b + b^3)*c)*f)*cosh(d*x + c)^2 + (a^2*b + b^3)*d*sinh(1)
- 2*((a^2*b + b^3)*d*cosh(1) + (a^2*b + b^3)*d*sinh(1) - (a^3 + a*b^2 + (a^2*b + b^3)*c)*f)*cosh(d*x + c)*sinh
(d*x + c) - ((a^2*b + b^3)*d*cosh(1) + (a^2*b + b^3)*d*sinh(1) - (a^3 + a*b^2 + (a^2*b + b^3)*c)*f)*sinh(d*x +
c)^2 - (a^3 + a*b^2 + (a^2*b + b^3)*c)*f)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + ((a^2*b + b^3)*d*f*x + (a^
2*b + b^3)*c*f - ((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*c*f)*cosh(d*x + c)^2 - 2*((a^2*b + b^3)*d*f*x + (a^2*b +
b^3)*c*f)*cosh(d*x + c)*sinh(d*x + c) - ((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*c*f)*sinh(d*x + c)^2)*log(-cosh(
d*x + c) - sinh(d*x + c) + 1))/((a^4 + a^2*b^2)*d^2*cosh(d*x + c)^2 + 2*(a^4 + a^2*b^2)*d^2*cosh(d*x + c)*sinh
(d*x + c) + (a^4 + a^2*b^2)*d^2*sinh(d*x + c)^2 - (a^4 + a^2*b^2)*d^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e + f x\right ) \operatorname {csch}^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*csch(d*x+c)**2/(a+b*sinh(d*x+c)),x)

[Out]

Integral((e + f*x)*csch(c + d*x)**2/(a + b*sinh(c + d*x)), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {e+f\,x}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e + f*x)/(sinh(c + d*x)^2*(a + b*sinh(c + d*x))),x)

[Out]

int((e + f*x)/(sinh(c + d*x)^2*(a + b*sinh(c + d*x))), x)

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